Exact Solution of the Wiener–Hopf Equation on the Segment for Contact Problems of the Theory of Cracks in a Layered Medium

This paper presents an approach that allows us for the first time to construct an exact solution of the Wiener–Hopf integral equations on a finite segment for the case of meromorphic functions in Fourier transforms of the kernel. The Wiener–Hopf integral equation is traditionally considered set on a semi-infinite segment. However, in applications, there are often cases of their application specified on a finite segment. For these purposes, approximate methods of applying these integral equations have been developed. However, when considering the Wiener–Hopf integral equations generated by mixed problems of continuum mechanics and mathematical physics in a multilayer medium of finite thickness, it turned out that these integral equations are solved exactly both on semi-infinite and finite segments. The approach is based on a new modeling method in differential equations and in some types of integral equations. It allows the reduction of Wiener–Hopf integral equations to infinite systems of linear algebraic equations that are solved exactly. The obtained result opens up the possibility of constructing exact solutions to boundary value problems for deformable stamps and cracks of a new type in bounded bodies.